Finite Group Databasebeta

← #26238281

Subgroups — #26238281

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689420C211yesmaximal, characteristic
2C4⋊C4×D188C211yesmaximal, characteristic
3#20244358C211yesmaximal, characteristic, complemented
4#20244151C211yesmaximal, characteristic, complemented
5#20244123C211yesmaximal, characteristic, complemented
6#20244113C211yesmaximal, characteristic, complemented
7#20243973C211yesmaximal, characteristic, complemented
8#20243959C211yesmaximal, characteristic, complemented
9#20243648C211yesmaximal, characteristic
10#20243648C211yesmaximal, characteristic
11#20243263C211yesmaximal, characteristic, complemented
12#20243216C211yesmaximal, characteristic
13#20243216C211yesmaximal, characteristic
14C4×#13093944C211yesmaximal, characteristic
15C47×#2370C211yesmaximal, characteristic, complemented
16C2×C4×D188C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×C4⋊C4×D47—21no—
20#15689420C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093944C2211yescharacteristic
29C2×#13093944C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32C2×C4×C47⋊C8C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15688997—21no—
36#15688997—21no—
37#15688993C2211yescharacteristic
38#15688980—21no—
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×#125×C47C2211yescharacteristic
50C47×#389—21no—
51C47×#389C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#387—21no—
54C47×#387—21no—
55C4×C4⋊C4×C47—21no—
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C4×D188—22no—
69C4×D188—22no—
70C4×D188—22no—
71C4×D188—41no—
72C4×D188—41no—
73C4×D188C2×C414yes—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C4⋊C4×D47—44no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85#13093957C2×C414yescomplemented
86#13093955—41no—
87#13093955—41no—
88#13093955C2×C414yescomplemented
89C4⋊(C47⋊C8)C2×C414yescomplemented
90#13093944—22no—
91#13093944C2×C414yes—
92#13093944C2×C414yes—
93#13093944Q814yes—
94#13093935—22no—
95#13093935—41no—
96C4×C47⋊C8—41no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99#125×C47—22no—
100#125×C47C2×C414yescomplemented
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C42×C47—21no—
112C2×C42×C47—21no—
113C2×C42×C47—21no—
114C2×C42×C47C2311yescharacteristic
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117#95×C47—21no—
118#95×C47—21no—
119#95×C47—21no—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47D411yescharacteristic
123#95×C47D411yescharacteristic
124C23×D47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188C22×C412yes—
143C2×D188C2×Q812yes—
144C2×D188#4112yes—
145C2×C4×D47—24no—
146C2×C4×D47—42no—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47—81no—
153C2×C4×D47—81no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—24no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—41no—
163C4⋊C4×C47—42no—
164C4⋊C4×C47—81no—
165C4⋊C4×C47C22×C412yes—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C2×Q812yes—
168C42×C47—21no—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—42no—
173C42×C47—81no—
174C42×C47—81no—
175C42×C47C22×C412yes—
176C42×C47#4112yes—
177(C2×D47)⋊C4—24no—
178(C2×D47)⋊C4—42no—
179(C2×D47)⋊C4—42no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—42no—
186C4⋊Dic47—81no—
187C4⋊Dic47C22×C412yes—
188Dic47⋊C4—42no—
189Dic47⋊C4—42no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8C22×C412yes—
200C2×C47⋊C8C22×C412yes—
201C2×C47⋊C8C2×Q812yes—
202C23×C47C22×D411yescharacteristic
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—41no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—22no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47C4×D412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47AGL(1,ℤ/8)12yes—
240C2×C4×C47C23×C411yescharacteristic
241C2×C4×C47C22×Q811yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47H(5,𝔽2)11yescharacteristic
245C2×C4×C472−1+411yescharacteristic
246C2×Dic47—22no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255D188—22no—
256D188—22no—
257D188—22no—
258D188C4×Q814yes—
259C4×D47—48no—
260C4×D47—84no—
261C4×D47—84no—
262C47⋊C8—22no—
263C47⋊C8—22no—
264C47⋊C8—22no—
265C47⋊C8—41no—
266#22978—471nomaximal, Sylow
267C22×C47C2×C4×D411yescharacteristic
268C22×C47#54611yescharacteristic
269C22×C47D4×Q811yescharacteristic
270C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C47#57511yescharacteristic
272C22×C47#57711yescharacteristic
273C22×C47#57711yescharacteristic
274C2×D47—24no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C2×D47—42no—
280C2×D47—42no—
281C4×C47—22no—
282C4×C47—22no—
283C4×C47—22no—
284C4×C47—42no—
285C4×C47—42no—
286C4×C47—42no—
287C4×C47—41no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—81no—
291C4×C47C2×C4×Q811yescharacteristic
292C4×C47#52011yescharacteristic
293C4×C47#55211yescharacteristic
294C4×C47#55411yescharacteristic
295Dic47—42no—
296Dic47—81no—
297Dic47—81no—
298C2×#478—471no—
299#2370D4711yescharacteristic, complemented
300D4×C4⋊C4—471no—
301#1992—471no—
302#1948—471no—
303#1943—471no—
304#1941—471no—
305#1920—471no—
306#1918—471no—
307#1864—471no—
308#1864—471no—
309#1771—471no—
310#1762—471no—
311#1762—471no—
312C4×#102—471no—
313C2×C47#236011yescharacteristic
314C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C47#295211yescharacteristic
316C2×C47#330711yescharacteristic
317C2×C47#331011yescharacteristic
318C2×C47#338411yescharacteristic
319C2×C47#338811yescharacteristic
320D47—44no—
321C2×#125C2×D4711yescharacteristic
322C2×C4×D4—471no—
323C2×C4×D4—471no—
324C2×C4×D4—471no—
325C22×C4⋊C4—941no—
326#478—478no—
327C2×#107—471no—
328C2×#106—471no—
329C2×#106—471no—
330C2×#106—471no—
331C2×C4⋊C8—471no—
332C2×#102—471no—
333C2×#102—471no—
334C2×#102—471no—
335C2×#102—471no—
336C2×#97—471no—
337C2×#97—471no—
338C2×C4×C8—471no—
339#389—21no—
340#389—471no—
341#389—471no—
342#389C2×D4711yescharacteristic
343#389C2×D4711yescharacteristic
344#387—21no—
345#387—21no—
346#385—941no—
347#385—941no—
348#384—471no—
349#380—941no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4C2×D4711yescharacteristic
352C4×C4⋊C4C2×D4711yescharacteristic
353C4×C4⋊C4C2×D4711yescharacteristic
354C4×C4⋊C4C2×D4711yescharacteristic
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#336—471no—
364#336—471no—
365C47#2297811yescharacteristic, complemented, Sylow
366C22×D4—471no—
367C23×C4—941no—
368C23×C4—941no—
369C23×C4—941no—
370C22×C8—471no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8—471no—
374#125—22no—
375#125C4×D4714yescomplemented
376C4×D4—474no—
377C4×D4—942no—
378C4×D4—942no—
379C4×D4—942no—
380C4×D4—1881no—
381C4×D4—1881no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—1884no—
394C2×C4⋊C4C22⋊D4711yescharacteristic
395C2×C4⋊C4C22⋊D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C22×D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4711yescharacteristic
409#107—474no—
410#106—474no—
411#106—1881no—
412#106—1881no—
413C4⋊C8—474no—
414#102—474no—
415#102—474no—
416#102—474no—
417#102—942no—
418#97—942no—
419#97—1881no—
420C4×C8—1881no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—941no—
427#95—941no—
428#95C22⋊D4711yescharacteristic
429#95C22⋊D4711yescharacteristic
430C24—941no—
431C2×D4—472no—
432C2×D4—472no—
433C2×D4—472no—
434C2×D4—941no—
435C2×D4—941no—
436C2×D4—941no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—941no—
442C22×C4—944no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—3761no—
446C22×C4—3761no—
447C22×C4—1882no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4—1882no—
451C22×C4—1882no—
452C22×C4—1882no—
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C2×C22⋊D4711yescharacteristic
456C22×C4C2×C22⋊D4711yescharacteristic
457C22×C4C2×C22⋊D4711yescharacteristic
458C22×C4C2×C22⋊D4711yescharacteristic
459C22×C4C23×D4711yescharacteristic
460C2×C8—472no—
461C2×C8—472no—
462C2×C8—472no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—472no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—3761no—
490C4⋊C4—1882no—
491C4⋊C4—1882no—
492C4⋊C4—1882no—
493C4⋊C4C2×C4×D4712yes—
494C4⋊C4C2×C4×D4712yes—
495C4⋊C4Q8×D4712yes—
496C22⋊C4—944no—
497C22⋊C4—1882no—
498C22⋊C4—1882no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—81no—
505C42—81no—
506C42C2×C4×D4712yes—
507C42#1062535012yes—
508C23—942no—
509C23—942no—
510C23—942no—
511C23—1881no—
512C23—1881no—
513C23—1881no—
514C23—1881no—
515C23C22×C22⋊D4711yescharacteristic
516D4—474no—
517D4—942no—
518D4—942no—
519D4—942no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—942no—
548C2×C4—3764no—
549C2×C4—3764no—
550C2×C4—1888no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4C4×C22⋊D4712yes—
560C2×C4C4×C22⋊D4712yes—
561C2×C4#1309404912yes—
562C2×C4C22×C4×D4711yescharacteristic
563C2×C4C2×Q8×D4711yescharacteristic
564C2×C4#41×D4711yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C2×C4#1309407011yescharacteristic
567C2×C4#1309407211yescharacteristic
568C8—942no—
569C8—942no—
570C8—942no—
571C8—1881no—
572C22—944no—
573C22—1882no—
574C22—1882no—
575C22—1882no—
576C22—1882no—
577C22—1882no—
578C22—1882no—
579C22C2×C4×C22⋊D4711yescharacteristic
580C22#1568976111yescharacteristic
581C22Q8×C22⋊D4711yescharacteristic
582C22C2×#1309404911yescharacteristic
583C22#1568992111yescharacteristic
584C22#1568993111yescharacteristic
585C22#1568993411yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—3761no—
597C4—3761no—
598C4—1882no—
599C4C4×Q8×D4711yescharacteristic
600C4#1568960511yescharacteristic
601C4#1568981411yescharacteristic
602C4#1568983311yescharacteristic
603C2—1884no—
604C2#2024638911yescharacteristic
605C2C4×#1309404911yescharacteristic
606C2#2025226911yescharacteristic
607C2#2025535811yescharacteristic
608C2#2025540311yescharacteristic
609C2#2025628711yescharacteristic
610C2#2025632711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2370C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#2370D47
hypocenterC47#22978
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20244358C2semidirect
4. #20244151C2semidirect
5. #20244123C2semidirect
6. #20244113C2semidirect
7. #20243973C2semidirect
8. #20243959C2semidirect
11. #20243263C2semidirect
15. C47×#2370C2semidirect
20. #15689420C4semidirect
85. #13093957C2×C4semidirect
88. #13093955C2×C4semidirect
89. C4⋊(C47⋊C8)C2×C4semidirect
100. #125×C47C2×C4semidirect
299. #2370D47semidirect
365. C47#22978semidirect
375. #125C4×D47semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156894202
C4⋊C4×D1882
#202443582
#202441512
#202441232
#202441132
#202439732
#202439592
#202436482
#202432632
#202432162
C4×#130939442
C47×#23702
#2297847